• A time series is a group of observations of a data point that are tracked over time
  • Trend/Cycle
  • Seasonal
  • Remainder
• This information can be tracked at several levels of time (e.g. annual, quarterly, monthly, etc.)
• Economic measures are generally the most common metrics tracked and discussed, but any area with relevant data points can be analyzed

• It is helpful to understand the patterns that make up trends/cycles vs seasonal behaviors.
• Different modeling techniques can be applied based on the knowledge of how the primary components interact with one another.
• If you are interested in making inferences on this data, it is a good starting point to get a better understanding of these elements

• We pulled data representing Citibike rides in Jersey City during Q4 of 2023

## # A tsibble: 6 x 5 [1h] <UTC>
## # Key:       start_station_name, rideable_type, member_casual [6]
## # Groups:    start_station_name, rideable_type @ start_time [6]
##   start_time          start_station_name rideable_type member_casual
##   <dttm>              <chr>              <chr>         <chr>        
## 1 2023-09-12 08:00:00 1 Ave & E 30 St    classic_bike  member       
## 2 2023-06-22 00:00:00 1 Ave & E 62 St    classic_bike  member       
## 3 2023-05-25 08:00:00 1 Ave & E 68 St    classic_bike  member       
## 4 2023-05-26 11:00:00 10 Ave & W 28 St   classic_bike  member       
## 5 2023-10-18 18:00:00 11 Ave & W 27 St   classic_bike  member       
## 6 2023-06-11 15:00:00 11 Ave & W 59 St   classic_bike  member       
## # ℹ 1 more variable: number_of_rides <int>

It’s always a good idea to plot your time series prior to any sort of decomposition

Below is the seasonal plot of our data for a daily period. We see a higher number of rides around 5-6pm, when many people are commuting.

• Lag plots can be an effective way to display autocorrelation and seasonal cycles within your TS

Below we see a plot of weekly lags. We don’t see a ton of structure here indicating weaker relationships between a value and its lagged values

• Our example time-series data has two seasonal periods: daily and weekly
• We can plot the different components of our decomposed series via the autoplot function

• We can also seasonally-adjust our data easily in the fabletools package

• Useful for “smoothing” data to reduce seasonal fluctuations and discover the trend-cycle.

\[ T_t = \sum_{j=-k}^{k} y_{t+j} \]

\[k = \frac {(m-1)}{2}\]

\[a_j = a_{-j}\]

\[7-MA = \sum_{j=-3}^{3} y_{t+j}\]

• Additive Decomposition: \(Y = S + T + C\)
  • Calculate Trend-Cycle Component \(T_t\) = 2 x m-MA (even) or m-MA (odd)
  • Detrended Series = \(y_t - T_t\)
  • Seasonal Series = Average of the seasonal component (month, quarter, etc)
  • Remainder (Irregular) Series = \(R_t = y_t - T_t - S_t\)

• Multiplicative: \(Y = S * T * C\)
  • Calculate Trend-Cycle Component \(T_t\) = 2 x m-MA (even) or m-MA (odd)
  • Detrended Series = \(\frac{y_t}{T_t}\)
  • Seasonal Series = Average of the seasonal component (month, quarter, etc)
  • Remainder (Irregular) Series = \(R_t = \frac{y_t}{T_t * S_t}\)

• The estimated trend-cycle is missing for the first and last few observations
• The trend cycle can be smoothed too much with large, fast peaks and valleys
• We assume the seasonality holds true between each year
• It is not robust to valid outliers such as natural disasters or economic changes.

• Seasonal and Trend Decomposition using LOESS
• Use trend and season parameters to control how rapidly seasonality and trend change

Pros

• Can handle different types of seasonality well (not just monthly/quarterly)
• User has a lot of control over parameters (smoothness, windowing, etc)
• Robust to outlier values

Cons

• Only can be used with additive decomposition
• Doesn’t handle calendar variation automatically
  • Here’s an article with a newer method that handles multiple seasonalities well!

Alternative Approaches

• Since economists spend most of their time analyzing time series data, they have come up with additional methods
  • X11/SEATS
    • Primarily handles monthly and quarterly data, but allows for incorporation of holidays.
    • Variation of the seasonal period is allowed
  • DSA
    • German Bank established this method to handle daily data
    • Combines STL on multiple seasonal components and ARIMA for additional handling of outliers and calendar adjustments

• Hyndman, R.J., & Athanasopoulos, G. (2021) Forecasting: principles and practice, 3rd edition, OTexts: Melbourne, Australia. OTexts.com/fpp3. Accessed on February 16th, 2024
• Oscar Trull, J. Carlos García-Díaz, A. Peiró-Signes, Multiple seasonal STL decomposition with discrete-interval moving seasonalities, Applied Mathematics and Computation, Volume 433, 2022, 127398, ISSN 0096-3003, https://doi.org/10.1016/j.amc.2022.127398
• https://cran.r-project.org/web/packages/dsa/dsa.pdf
• https://towardsdatascience.com/seasonal-adjustment-of-daily-time-series-1bd2aa9b096d